Using Kolmogorov Complexity with Graph and Vertex Entropy to Measure Similarity of Empirical Graphs with Theoretical Graph Models

Abstract

Over the years, several theoretical graph generation models have been proposed. Among the most prominent are: Erd\H{o}s-Renyi random graph model, Watts-Strogatz small world model, Albert-Barab\'{a}si preferential attachment model, Price citation model, and many more. Often, researchers working on an empirical graph want to know, which of the theoretical graph generation models is the closest, i.e., which theoretical model would generate a graph the most similar to the given empirical graph. Usually, in order to assess the similarity of two graphs, centrality measure distributions are compared. For a theoretical graph model this commonly means comparing the empirical graph to a realization of the theoretical graph model, where the realization is generated from the given model using arbitrarily set parameters. The similarity between centrality measure distributions can be measured using standard statistical tests, e.g., the Kolmogorov-Smirnov test of distances between cumulative distributions. This approach is both error-prone and leading to incorrect conclusions. In this work we present a general framework for comparing graphs with theoretical models. Our framework is twofold. First, we show how comparing the entropies of centrality measure distributions (degree centrality, betweenness centrality, closeness centrality, eigenvector centrality) can help assign an empirical graph to the most similar theoretical model. Second, we compare graphs with theoretical graph models based on the perceived complexity of graphs, which in turn is computed as the Kolmogorov Complexity (also known as the algorithmic entropy) of the graph's adjacency matrix. As the result, we introduce a robust and efficient method of assigning an empirical graph to the most probable theoretical graph model.